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Scalars and Vectors Study Guide (page 2)

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Updated on Sep 26, 2011

Example 1

The following Cartesian coordinates characterize a point: x = 7.0 and y = 2.0. Find the polar coordinates of this point.

Solution 1

Considering the previous explanation for polar coordinates, we can first find the distance to the point in polar coordinates, r, and then, applying trigonometric functions, we can determine the angle θ.

    x = 7.0
    y = –2.0
    r = ?
    θ = ?
    r2 = x2 + y2
    r2 = 72 + (–2)2
    r2 = 53
    r = 7.3
    sin θ = – 2.0/7.3
    θ = –16°

Both these two-dimensional systems of coordinates may be extended to three dimensions by adding a z-axis perpendicular to the plane, and we get the 3-D Cartesian and, respectively, the cylindrical coordinates systems.

For a 3-D Cartesian system of reference, the distance r of point P(x0, y0, z0) having coordinates x0, y0, and z0 to the origin is given by:

To go from this to the 3-D cylindrical system, we use the same relationship between (x, y) and (r, θ) in the horizontal plane, and the height z is same: (r,θ,z).

Example 2

Find the cylindrical system coordinates of point Q of (Cartesian) coordinates (3,4,6) units.

Solution 2

We have:

r2 = x2 + y2

r2 = 32 + 42 = 25

and therefore:

r = 5

sin θ = component on y/r

sin θ = 4/5 = 0.8

and therefore:

θ = 53°

Point Q coordinates in the cylindrical system are (5,53°,6).

Scalars and Vectors

Each of the physical quantities we may encounter in physics can be categorized as either scalars or vectors. A quantity that is completely specified only by its magnitude using a certain unit is called a scalar quantity. When you say, "I bought 3 lbs of tomatoes: " you have pretty much specified all the relevant information about your purchase. Therefore, mass is a scalar quantity. Other examples of scalar quantities include temperature, volume, pressure, density, and time. When you say that the temperature outside is 55° F, you do not need to specify a direction. Scalar quantities can be used in regular arithmetic computations with the only precaution being to express them using the same units.

At other times, in order to fully characterize a physical quantity, we need to specify both a magnitude and a direction. For example, in order to get from Lansing, Michigan, to Detroit, Michigan, you have to travel approximately 70 miles due east. Going in the wrong direction will take you to Grand Rapids or maybe Jackson, instead of Detroit.

A vector quantity therefore has both a magnitude and a direction. In the previous example, we say that displacement is a vector. We usually draw a vector as an arrow. The direction of the arrow gives the direction of the vector, and its length is usually scaled proportionally to the magnitude of the vector, using appropriate units. The magnitude of a vector is always a positive number. A vector symbol is a bold letter, for instance, A.

For a 3-D Cartesian system, we define a unit vector along each one of the axes; we do so by making a vector along the axis, pointing it in the positive direction, and making it have a unit length. The usual notation is i for the unit vector along the x-axis, j for the unit vector along the y-axis, and k for the unit vector along the z-axis.

| i | = | j | = | k | = 1

Unit vectors are used to specify the directions of the reference axes. We also define a vector's components Ax, Ay, and Az as the difference of its starting point and end tip coordinates (see Figure 2.4).

Scalars and Vectors

Then, we may represent the vector as:

A = i Ax + j Ay + k Az

Assuming the starting point of vector A is P(xs, ys, zs) and the end tip is at point Q(xt, yt, zt) then:

Ax = xtxs Ay = ytys Az = ztzs

The magnitude of a vector is:

A2 = Ax2 + Ay2 + Az2

Vector Properties

Vectors enjoy certain properties that can make their handling easier. For example, the components of any vector can be used in place of the vector itself in any calculation where it is convenient to do so.

Two vectors are equal if they have the same magnitude and the same direction. As the direction is specified by a whole bundle of parallel lines to each other, this allows us most of the time to move a vector parallel to itself without affecting the outcome of the problem.

Example 1

Vector Properties

All three vectors A, B, and C in Figure 2.5 are equal to one another. Their directions are parallel, and their magnitudes are equal.

We define the sum of two vectors as a new vector, also called a resultant vector.

One method to obtain the resultant vector is by moving one of the vectors parallel to itself until its starting point coincides with the end of the first one. The resultant vector starts at the tail of the first vector and ends at the end tip of the second vector.

We write the resultant vector as:

R = A + B

Without any loss of generality, we can limit ourselves to the two-dimensional case of vectors in the same plane.

Example 2

Show graphically the addition of two vectors A and B.

Solution 2

As defined previously, we move the second vector parallel to itself so that its tail intersects the tip of the first

 

Vector Properties

vector (see Figure 2.6). If the two vectors being added have components (Ax, Ay, Az) and (Bx, By, Bz), the sum or resultant can be easily seen to have the components (Rx, Ry, Rz) given by:

Rx = Ax + Bx

Ry = Ay + By

Rz = Az + Bz

and the magnitude of the resultant is:

R2 = Rx2 + Ry2 + Rz2

While the resultant vector can be written as:

R = i · Rx + j · Ry + k · Rz

The opposite of a vector A is a vector – A, having the same magnitude, but oriented in the opposite sense relative to the initial vector (see Figure 2.7).

Vector Properties

Vector Properties

This allows us to define the difference of two vectors by means of adding one of the vectors with the opposite of the other.

AB = A + (–B)

In Figure 2.8, the resultant vector of AA = A + (–A) = 0 is demonstrated. The two vectors have intentionally not been perfectly superimposed as an aid to the eye.

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